Majorization and Semi-Doubly Stochastic Operators on $L^1(X)$

TL;DR

This paper explores majorization in $L^1(X)$ using semi-doubly stochastic operators, providing characterizations, relations to integral stochastic operators, and compactness properties.

Contribution

It characterizes majorization via semi-doubly stochastic maps on $L^1(X)$ and answers Mirsky's question, advancing the understanding of these operators.

Findings

Characterization of majorization through semi-doubly stochastic operators.

Established the relation between semi-doubly stochastic and integral stochastic operators.

Proved relative weak compactness of certain operator sets via equi-integrability.

Abstract

This article is devoted to a study of majorization based on semi-doubly stochastic operators (denoted by $S\mathcal{D}(L^1)$) on $L^1(X)$ when $X$ is a $\sigma$-finite measure space. We answered Mirsky's question and characterized the majorization by means of semi-doubly stochastic maps on $L^1(X)$. We collect some results of semi-doubly stochastic operators such as a strong relation of semi-doubly stochastic operators and integral stochastic operators, and relatively weakly compactness of $S_f=\{Sf: ~S\in S\mathcal{D}(L^1)\}$ when $f$ is a fixed element in $L^1(X)$ by proving equi-integrability of $S_f$.